In 2011, a U.S. Census report determined that 71\% of college students are working students. A researcher thinks this percentage has changed and surveys 186 college students. The researcher reports that 123 of the 186 are working students. Is there evidence to support the researcher's claim at the $1 \%$ significance level?
a. Determine the null and alternative hypotheses.
Answer
Final Answer: There is not enough evidence to support the researcher's claim that the proportion of working students has changed at the 1% significance level. Therefore, the answer is \(\boxed{\text{No}}\).
Step 1 ：Define the null and alternative hypotheses. The null hypothesis is that the proportion of working students is still 71%, while the alternative hypothesis is that the proportion has changed. We can use a two-tailed test to determine if there is a significant difference.
Step 2 ：Calculate the z-score and compare it to the critical value for a 1% significance level. If the z-score is greater than the critical value, we reject the null hypothesis.
Step 3 ：Given values: \(p0 = 0.71\), \(p = 0.6612903225806451\), \(n = 186\), \(SE = 0.033271447929295594\), \(z = -1.4640083450190289\), \(critical\_value = 2.5758293035489004\)
Step 4 ：The z-score is less than the critical value, so we fail to reject the null hypothesis. This means that there is not enough evidence to support the researcher's claim that the proportion of working students has changed at the 1% significance level.
Step 5 ：Final Answer: There is not enough evidence to support the researcher's claim that the proportion of working students has changed at the 1% significance level. Therefore, the answer is \(\boxed{\text{No}}\).
